Theorem 0nnq 10882

Description: The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
0nnq	⊢ ¬ ∅ ∈ Q

Proof of Theorem 0nnq

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0nelxp 5681	. 2 ⊢ ¬ ∅ ∈ (N × N)
2		df-nq 10870	. . . 4 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))}
3	2	ssrab3 4035	. . 3 ⊢ Q ⊆ (N × N)
4	3	sseli 3932	. 2 ⊢ (∅ ∈ Q → ∅ ∈ (N × N))
5	1, 4	mto 199	1 ⊢ ¬ ∅ ∈ Q

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2142  ∀wral 3076  ∅c0 4285   class class class wbr 5100   × cxp 5645  ‘cfv 6521  2^nd c2nd 7969  Ncnpi 10802   <_N clti 10805   ~_Q ceq 10809  Qcnq 10810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-opab 5163  df-xp 5653  df-nq 10870

This theorem is referenced by:  adderpq  10914  mulerpq  10915  addassnq  10916  mulassnq  10917  distrnq  10919  recmulnq  10922  recclnq  10924  ltanq  10929  ltmnq  10930  ltexnq  10933  nsmallnq  10935  ltbtwnnq  10936  ltrnq  10937  prlem934  10991  ltaddpr  10992  ltexprlem2  10995  ltexprlem3  10996  ltexprlem4  10997  ltexprlem6  10999  ltexprlem7  11000  prlem936  11005  reclem2pr  11006